例6证明级数∑ 是收敛的,并求和 n1n(n+1)(n+2) 解 nn+Xn+2)2(n+)(n+(m+2) (i+1i+2) 111 2(12232334n(n+1)(n+1)(n+2) 2(2(n+1)(n+2)4例6 证明级数 是收敛的，并求和。 1 1 n n n( 1 ) ( n 2 ) ∞ = + + ∑ 解 ( ) ( ) 1 1 1 1 n n( 1 ) ( n 2) 2 n ( n 1 ) n 1 n 2 ⎛ ⎞ = − ⎜ ⎟ ⎜ ⎟ + + + + + ⎝ ⎠ ∵ ( ) ( ) ( ) ( ) ( ) 1 1 ( 1 ) ( 2 ) 1 1 1 1 1 1 1 2 1 2 2 3 2 3 3 4 1 1 2 1 1 1 1 . 2 2 1 2 4 n i i i i n n n n n n = + + ⎛ ⎞ = − ⎜ ⎟ + − + + − ⎜ ⎟ ⋅ ⋅ ⋅ ⋅ ⋅ + + + ⎝ ⎠ ⎛ ⎞ = + ⎜ ⎟ → ⎜ ⎟ + + ⎝ ⎠ ∴ ∑